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> But, at most countably many of those sets will ever be individually thought about by any human being. If your arguments are tied to such physical constraints, then there aren't even countably infinite natural numbers unless the Universe is infinite in space or in time (such that there can be infinite humans or "thinkers").
Indeed, that's why I'm an ultrafinitist. https://en.wikipedia.org/wiki/Ultrafinitism
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This is discussed heavily in topics around constructing the reals and constructivism. Wikipedia has a short section about using the computables instead of the reals: https://en.m.wikipedia.org/wiki/Computable_number It is pretty fascinating to think about: that almost all real numbers are not computable ("almost all" of course meaning "all but a countable set"). And yet you almost certainly will never run into a nonc…
Are there any good books on this topic?
(actually - I'm guessing there's actually a countable number in the Library of Babel but it didn't read quite so amusingly that way. In any case - all the ones I flicked through were trash.)
Edit - The Library of Babel is actually finite isn't it? Fixed alphabet and fixed book length? It's a while since I read it.
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This is discussed heavily in topics around constructing the reals and constructivism. Wikipedia has a short section about using the computables instead of the reals: https://en.m.wikipedia.org/wiki/Computable_number It is pretty fascinating to think about: that almost all real numbers are not computable ("almost all" of course meaning "all but a countable set"). And yet you almost certainly will never run into a nonc…
Are there any good books on this topic?
[0]: https://www.amazon.com/Meta-Math-Quest-Gregory-Chaitin/dp/14...
- the number of prisoners is infinite, so they will never finish answering the question. At any point in time, only a finite number of prisoners will be freed.
- a single prisoner must process an infinite amount of information to reach the decision. In fact, by observing only a finite number of hats he cannot possibly choose the answer.
- the number of equivalence classes is uncountable. Not even a countably infinite number of prisoners can possibly have enough time to pick out a single element from every equivalence class.
As time goes by, I increasingly view things like uncountable infinities and the axiom of choice as "a fun maths game", rather than having any intrinsic truth or falsity. Other's view may differ. Here is an interesting thing I've never seen anyone write down (I should do it myself) -- we don't need uncountable infinities. * How many natural numbers are there? Countable * How many rational numbers are there? Countable…
Infinities aren't real, so you shouldn't be surprised if unrealistic things happen when you invoke infinities. That you can duplicate a sphere by cutting it into a finite number of pieces and reassembling it is a "fact" in the same sense as "Luke Skywalker destroyed the Death Star". It might be interesting and culturally important, but it's talking about fictional entities. Both spheres and arbitrarily detailed piece…
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> Don't they just say that if there's such a thing as objective mathematical reality, it can't be effectively axiomatized? No they don't. Real space doesn't appear to be infinite, and Zn is not subject to Godel's incompleteness theorem. If you drop the requirement of infinite numbers and "recursive" infinites (e.g. real numbers), as reality appears to do, there is no problem.
Ok. Not sure I buy that (eg right now, we have no reason to believe that the lifetime of the universe is finite), but that's not what I was getting at: My point is that at worst, the incompleteness theorems only imply that we won't be able to write down all the rules that govern the universe.
Not being able to write all the rules is a completely different and unrelated thing.
There are many problems with this puzzle that go against the intuition. - the number of prisoners is infinite, so they will never finish answering the question. At any point in time, only a finite number of prisoners will be freed. - a single prisoner must process an infinite amount of information to reach the decision. In fact, by observing only a finite number of hats he cannot possibly choose the answer. - the num…
How do I guess the color of my hat if there's an uncountable number of possible colors? Doesn't that mean that communicating the value of that color involves transmitting an infinite amount of informtion?